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Breaking the imaging symmetry in negative
refraction lenses
Changbao Ma and Zhaowei Liu*
Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, California
92093-0407, USA
*
[email protected]
Abstract: Optical lenses are pervasive in various areas of sciences and
technologies. It is well known that conventional lenses have symmetrical
imaging properties along forward and backward directions. In this letter, we
show that hyperbolic plasmonic metamaterial based negative refraction
lenses perform as either converging lenses or diverging lenses depending on
the illumination directions. New imaging equations and properties that are
different from those of all the existing optical lenses are also presented.
These new imaging properties, including symmetry breaking as well as the
super resolving power, significantly expand the horizon of imaging optics
and optical system design.
© 2012 Optical Society of America
OCIS codes: (160.3918) Metamaterials; (220.3630) Lenses; (110.2990) Image formation
theory; (350.4238) Nanophotonics and photonic crystals.
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Received 23 Nov 2011; revised 13 Jan 2012; accepted 13 Jan 2012; published 20 Jan 2012
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1. Introduction
Lenses are the most fundamental and widely used elements in optics. The configuration of a
lens is determined by the type of wavefront conversion to perform; basic conversions include
transformation among diverging waves, converging waves and plane waves. As an object can
be considered as an accumulation of point sources, a lens that converts a diverging wave to a
converging one is capable to form an image of the object.
Imaging is one of the most basic functions of a lens, of which the imaging properties have
been well established since the first elaboration by Kepler in 1611 [1]. As is illustrated in Fig.
1 and summarized in Table 1 [2], the image shows various characteristics when an object is
placed at different locations with respect to the focal length f of a lens. Such imaging rules,
which are governed by the imaging Eq. (1)/s0 + 1/si = 1/f with s0 and si being the object and
image distances, respectively, are the foundations for ray optics and optical system design.
Fig. 1. Imaging behaviors of a conventional converging lens. An object is placed beyond 2f in
(a), at 2f in (b), between f and 2f in (c), and within f in (d). f is the focal length of the lens.
Table 1. Imaging Properties of a Conventional Converging Lens
Object
Location
∞>s>2f
s = 2f
f<s<2f
s=f
s<f
Type
Real
Real
Real
Virtual
Location
f<p<2f
p = 2f
2f<p<∞
±∞
-∞<p<0
Image
Orientation
Inverted
Inverted
Inverted
Relative size
Minified
Same size
Magnified
Erect
Magnified
Based on the basic properties of a single lens, systems comprising multiple lenses can be
conveniently designed. The resolution of an optical system is however beyond the scope of
ray optics and has to be examined by wave optics. Owing to the diffractive nature of light, the
best resolution of a lens system is limited to about half of the working wavelength ~λ/2 [3].
Using the immersion technique, the resolution can be improved up to ~λ/(2n), but such
enhancement however is quite modest due to the low refractive index n of natural materials
[4, 5]. To achieve higher resolution, a material that can support the propagation of light with
higher k-vectors is needed. Metamaterials, which are artificially engineered nanocomposites
possessing extraordinary material properties, provide new opportunities for this purpose [6–
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Received 23 Nov 2011; revised 13 Jan 2012; accepted 13 Jan 2012; published 20 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 2582
8]. Various metamaterial based lenses have been proposed and demonstrated within the last
decade [9–21]. Despite their success in achieving super resolution, such superlenses behave
distinctively to their conventional counterpart. For instance, they cannot focus a plane wave,
which represents a significant defect from both theoretical and practical points of view.
2. Theory
By introducing various phase compensation mechanisms in metamaterials, the plane wave
focusing function has been realized recently [22, 23]. In this paper, we first provide a
generalized imaging equation for the phase compensated metamaterials lenses, in which focal
lengths can be defined from either direction. We further illustrate the unprecedented imaging
properties if such lenses also experience negative refraction at the lens/air interfaces.
Fig. 2. Focusing behaviors of a hyperbolic metalens. (a) Plane wave from air converges to a
focus (Fm) in metamaterial. (c) Plane wave from metamaterial diverges in air, resulting in a
virtual focus (Fd) also in metamaterial. The arrows in (a) and (c) are eye-guiding rays. (b) and
(d) Symbolized representation of (a) and (c). The dashed vertical lines represent phase
compensating elements at the interfaces, i.e. the PWC.
By convenience, we use the hyperbolic metalens, which comprises a metal plasmonic
waveguide coupler (PWC) (gray structure in Fig. 2(a) and 2(c)) and plasmonic metamaterials
with hyperbolic dispersion (εx’ >0, εz’ <0), as a specific example to illustrate the new imaging
paradigm. Such a hyperbolic metalens can be designed to focus an incident p-polarized plane
wave from the air side to the focus Fm at the focal length fm in the metamaterial by given εx’
and εz’. The details of the PWC design follows similar principles described in literatures [23–
25]. Due to the hyperbolic dispersion of the metalens, a plane wave comes from the
metamaterial side, instead of being focused, diverges after the metalens, resulting in a virtual
focus at Fd also in the metamaterial (Fig. 2(c)). It is well known that a conventional
converging lens has two foci: one on each side; i.e., a plane wave coming from one side
focuses to the other, and vice versa. In stark contrast, a hyperbolic metalens works like a
converging lens from one side but a diverging lens from the other. Similar to a recent “Janus
device” [26] showing different functions along multiple directions based on transformation
optics [27, 28], a hyperbolic metalens is a “Janus lens” having two different focusing
behaviors in opposite directions. Interestingly, although both focal lengths can be clearly
defined, different signs must be taken accordingly. This distinctive focusing behavior leads to
extraordinary imaging properties that do not exist in common lenses.
3. Formulations and simulations
An image Pd in air is formed by an object Pm in the metamaterial, and vice versa, as
schematically shown in Fig. 3(a). The optical path length (OPL) from Pd to Pm is given
by (ϕd + ϕ m + ϕc ) / k0 , where φd = k0 vd2 + x 2 is the phase delay in air, φm = k0 ε x' vm2 + ε z' x 2 is
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the phase delay in the metamaterial, and φc = φconst − k0 ε x' f m2 + ε z' x 2 is the designed phase
delay profile of the PWC according to the metalens design principle. εx′ and εz′ are the real
part of the metamaterial permittivity in x and z directions respectively. vd and vm are the
image/object distances in air and metamaterial, and φconst is a constant.
According to the Fermat’s Principle, the OPL remains stationary; that is, its derivative
with respect to the position variable x is zero, i.e., d[OPL]/dx = 0. So we have
1
vd2 + x 2
+
ε z'
ε x' vm2 + ε z' x 2
=
ε z'
ε x' f m2 + ε z' x 2
(1)
This relationship must hold true for all the rays from Pm to Pd. Considering the paraxial
approximation (x is negligible), Eq. (1) can be reduced to
'
'
'
'
1 εz / εx εz / εx
+
=
vd
vm
fm
(2)
It can be seen from Eq. (2) that when vm → ∞ , i.e., the incident beam is a plane wave
from the metamaterial, the focal length in air is resulted as f d = f m ε x' / ε z' . Replacing fd into
Eq. (2), we obtain another form of metalens imaging equation
'
'
1 εz / εx
1
+
=
(3)
vd
vm
fd
Because of the hyperbolic dispersion of the metamaterial, the relation between fd and fm
indicates they always have different signs. According to the sign convention, if fm>0
represents Fm in the metamaterial (Fig. 2 (a)), fd<0 means that the Fd is not in air but also in
the metamaterial. Through an analogy to the typical symbolic notation of a focusing lens by
using a double arrowed line, we symbolize the hyperbolic metalens with both foci in the
metamaterial, as shown in Fig. 2(b) and 2(d).
(b)
(a)
(c)
fm
Pd
x
vm
vd
Air
Pm
z
Metamaterial
Fig. 3. (a) Schematic representation of forming an image by a hyperbolic metalens. Imaging
behavior of a hyperbolic metalens for an object in air: (b) Ray diagram, (b) Numerical result.
As is analyzed above, a hyperbolic metalens breaks the forward/backward symmetry
preserved in conventional lenses, leading to exceptional focusing behaviors. In the following,
we examine the imaging formation for an object placed at various locations with respect to its
focal lengths. The specific hyperbolic metalens was designed with fm = 2.0 µm at the
wavelength of 690 nm. The metamaterial is assumed to have εx = 6.4 + 0.03i and εz = −10.3 +
0.2i. This specific metamaterial can be constructed by silver nanowires in a dielectric host
with a refractive index of 1.3. The silver nanowires are orientated in the z direction and their
volumetric filling ratio is 0.5.
Figure 3(a) schematically depicts the imaging behavior of such a hyperbolic metalens for
an object in air. Two characteristic rays can be used to determine the image: the one parallel
to the optical axis is “refracted” toward the focus Fm in the metamaterial; another one towards
the focus Fd is “refracted” to the direction parallel to the optical axis in the metamaterial. The
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Received 23 Nov 2011; revised 13 Jan 2012; accepted 13 Jan 2012; published 20 Jan 2012
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crossing point of these two rays determines the location of the image. Particularly, in the case
of an object in air, the image in the metamaterial is always minified, erect, real and within the
first focal length, regardless of the distance of the object in air to the metalens. This is
completely different from a conventional lens, in which the image properties are dependent on
the position of the object (see Table 1). Another special ray, which is the one passing the
optical center O, may also be used to determine the location of the image through the
calculation of its refraction angle in the metamaterial θ m ≈ Re( ε x / ε z ) sin θ d , with θ d being
the incident angle and θ m being the refracted angle of that ray with respect to the optical axis.
Figure 3(b) shows the numerical verification for such a case using full wave simulation.
Because the image cannot go beyond the focal plane in the metamaterial, the entire half space
in air is mapped into a highly confined region in metamaterial.
Fig. 4. Imaging behavior of a hyperbolic metalens for an object in metamaterial. (a) Object is
inside point Fm. (c) Object is between points Fm and 2Fm. (e) Object is at point 2Fm. (g) Object
is outside point 2Fm. (b), (d), (f), (h), Numerical verifications of (a), (c), (e), (g), respectively.
Similarly, through schematic and geometric analysis, the image of an object in
metamaterial can also be analyzed: the property of the image is dependent on the position of
the object relative to the focus fm. The image location can also be determined using
characteristic rays: the one connecting the object and the focus Fm is “refracted” to the
direction parallel to the optical axis; another one parallel to the optical axis is “refracted” to
the direction whose backward extension passes the focus Fd, because Fd is not in air but in the
metamaterial. Figure 4(a) shows the imaging behavior of an object within the first focal length
in the metamaterial, i.e., vm < fm: the image in air is always magnified, erect and real. The
image is real because the “refracted” rays converge in air. This case is simply the reciprocal of
the case shown in Fig. 3(b). When the object is outside the first focal length, i.e., vm > fm the
rays in air diverge, but their backward extensions converge into a point on the other side of
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Received 23 Nov 2011; revised 13 Jan 2012; accepted 13 Jan 2012; published 20 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 2585
the optical axis in the metamaterial space, thus the image is always virtual and inverted.
Depending on the relative position of the object, the image may be different in size; when the
object is outside the first but inside the second focal length, i.e., fm < vm < 2fm, the image is
magnified, as shown in Fig. 4(c); when the object is at the second focal length, i.e., vm = 2fm,
the image has the same size as the object, as shown in Fig. 4(e); when the object is outside the
second focal length, i.e., vm > 2fm, the image is minified, as shown in Fig. 4(g). All the four
cases above for an object in the metamaterial are numerically verified, as shown in Fig. 4(b)(h). These extraordinary properties of a hyperbolic metalens are summarized in Table 2,
which can also be directly derived from the metalens imaging equations (Eq. (2) and (3)).
Table 2. Imaging Properties of a Hyperbolic Metalens. fm>0 and fd < 0.
Object
Location
∞>vd>0
∞>vm>2fm
vm = 2fm
fm<vm<2fm
vm = f m
vm<fm
Type
Real
Virtual
Virtual
Virtual
Real
Location
0<vm<fm
2fd<vd<fd
vd = 2fd
-∞<vd<2fd
±∞
0<vd<∞
Image
Orientation
Erect
Inverted
Inverted
Inverted
Relative size
Minified
Minified
Same size
Magnified
Erect
Magnified
As shown in Fig. 3 and Fig. 4, the magnification of the metalens can also be determined
analytically through the transverse magnification MT
MT =
um
g
f
=− m =− d
ud
fm
gd
(4)
where um and ud are the object/image height, i.e., the distance from the object/image to the
optical axis, in the metamaterial and air space respectively; gm and gd are the object/image
distance reckoned from their focus in the metamaterial and air space respectively.
4. Discussions and conclusions
Although illustrated using hyperbolic metalenses with fm>0, fm can also be designed to be
negative (fm<0), meaning a plane wave from the air side diverges in the hyperbolically
dispersive metamaterial. In this case, a plane wave from the metamaterial side will be focused
in air, resulting in a positive fd. Analyses show that all the above equations remain valid and
the hyperbolic metalens with fm<0 is also a “Janus lens”. The imaging properties of such a
hyperbolic metalens with fm<0 can also be deduced through a similar method mentioned
above.
We finally emphasize that the new set of imaging properties in the exampled hyperbolic
metalens is a result of the hyperbolic dispersion of the metamaterials. The hyperbolic
metalenses also possess super resolution capabilities as demonstrated before [23, 24]. The
rules are applicable to any other type of phase compensated lenses that experiences negative
refraction at the air/lens interface. These exotic imaging properties are impossible to realize
by conventional lenses, thus extending the imaging properties and the capabilities of lenses,
and the imaging optics in general, to a new horizon. A practical usage of such lenses is to
truncate the metamaterial to the focal/image plane so that the optical near field can be
accessed from air [29]. The new found imaging behaviors and the super resolving power of
hyperbolic metalenses provide new opportunities to explore novel optical devices and systems
and will profoundly affect the advancement of optics.
Acknowledgment
This work was partially supported by the startup fund of the University of California at San
Diego and the NSF-ECCS under Grant No. 0969405.
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Received 23 Nov 2011; revised 13 Jan 2012; accepted 13 Jan 2012; published 20 Jan 2012
30 January 2012 / Vol. 20, No. 3 / OPTICS EXPRESS 2586